Unformatted text preview: y) if 0=2 y < <=1
2 if 1
y1
2(1
if 1
y u = x sin 8 x cos y sin ;1 y
v = ; y cos 8 sin y:
Here u, v, and w are the Cartesian velocity components and = 5=3, = 6, and x and
y were chosen to be small. The boundary conditions specify that u = 0 on the sides and
top and v = 0 on the bottom.
Solutions for the density at t = 1:8 are shown in Figure 10.3 for computations
with p = 0 to 3. The mesh used for all values of p is shown in Figure 10.3. The total
number of vector degrees of freedom for twodimensional discontinuous Galerkin methods
is N np. Since there are four unknowns per element ( , m, n, and e) for twodimensional
ows, there are 2016, 6048, 12096, and 20160 unknowns for degrees p = 0, 1, 2, and
3, respectively. Fluxes were evaluated using Roe's linearized ux approximation 23].
No limiting was used for this computation. A highfrequency ltering 22] was used to
suppress oscillations in the vicinity of the interface separating the two uids. 10.3. Multidimensional Discontinuous Galerkin Methods 37 1/4 ρ=2 1/2 ρ=1 1/2 Figure 10.3.3: Con guration for the RayleighTaylor instability of Example 10.3.1. There
are solid walls on the bottom and sides and open ow at the top.
The results with p = 0 show very little structure of the solution. Those with p = 1
show moreandmore detail of the ow. There is no exact solution of this problem, so
it is not possible to appraise the e ects of using higher degree polynomials however,
solutions with more detail are assumed to be more correct.
Remacle et al. 22] also did computations using adaptive pre nement. There is no
error estimate available for the Euler equations, so they used an error indicator Ej on
element j consisting of
Z
3
XZ
Ej = r r dV +
( j ; nb )dS :
j k=1 @ jk j This can be shown 22] to be the length of the interface that separates the two uids
on j . Remacle et al. 22] increased the degree on elements where Ej was above the
median of all error indicators. Results using this adaptive pre nement strategy with p
ranging from 1 to 3...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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